Robust Fault Detection and Isolation for Stochastic ... - Semantic Scholar
Robust Fault Detection and Isolation for Stochastic Systems Jemin George and Irene M. Gregory Abstract—This paper outlines the formulation of a robust fault detection and isolation scheme that can precisely detect and isolate simultaneous actuator and sensor faults for uncertain linear stochastic systems. The given robust fault detection scheme based on the discontinuous robust observer approach would be able to distinguish between model uncertainties and actuator failures and therefore eliminate the problem of false alarms. Since the proposed approach involves precise reconstruction of sensor faults, it can also be used for sensor fault identification and the reconstruction of true outputs from faulty sensor outputs. Simulation results presented here validate the effectiveness of the robust fault detection and isolation system.
I. I NTRODUCTION Faults are deviations from the normal behavior of the plant or its instrumentation and they can be categorized into: i) additive process faults, ii) multiplicative process faults, iii) sensor faults, and iv) actuator faults. There exist several fault monitoring procedures which can be used to recognize and distinguish different types of faults [1]. These fault monitoring procedures can be categorized into: i) fault detection, ii) fault isolation, and iii) fault identification. A survey on design methods for fault detection is given in [2]. Most of the existing FDI (fault detection and isolation) schemes are based on measurement residual generation. Generated residual is used to facilitate the decision making procedures involved in FDI. The basic difference between most FDI schemes is the underlying residual generation methods. Few examples of different FDI schemes are the observer based Fault Detection Filters [3]–[5], Kalman filter based Proportional Integral Observers [6], Multiple Model Adaptive Estimators [7], and system identification methods [8]. In this manuscript two types of faults are of concern, i.e., actuator faults and sensor faults. The FDI scheme considered here is the observer based approach. Discontinuous observers such as the sliding mode observers have been successfully used in FDI context [9]. Design of sliding mode observers for detection and reconstruction of actuator and sensor faults is presented in [10] and [11], respectively. The precise reconstruction of faults proposed in [10] assumes the absence of uncertainty. The FDI approaches presented in [11] and [12] introduce an approximate fault reconstruction scheme by minimizing the error between the true fault signal and the reconstructed fault signal. In [13], an FDI scheme for a class of nonlinear uncertain systems is presented by introducing limitations on the structure of the uncertainty. It is important to notice that the precise reconstruction of fault is generally not available in the presence of uncertainty. While the FDI schemes presented in [10]–[13] involves reconstruction of faults, sliding mode observer based FDI approach presented in [14] is based on the measurement residual generation. Though the FDI scheme presented in [14] assumes precise knowledge of system dynamics, a similar FDI scheme which is robust to mismatched uncertainties is presented in [15]. The residual generation presented in [14] and [15] is based on the sliding mode observer scheme where the observer maintain the sliding motion in the presence of mismatched uncertainties, but when fault occurs, the sliding motion is broken and a residual is generated which contains information regarding the fault. J. George is with Dept. of Mechanical & Aerospace Engineering, University at Buffalo, SUNY, Buffalo, NY-14260 [email protected] I. M. Gregory is with NASA Langley Research Center, Hampton, VA-23681
[email protected]
Though there are numerous literatures on fault detection and identification, very little work is done on developing robust fault detection schemes. Robust actuator/sensor fault detection is a challenging problem due to system modeling errors, external disturbances, and measurement noise. Most of the current fault detection algorithms are model based and they tent to induce false alarms when the plant dynamics differ from the assumed model. This manuscript presents a robust FDI scheme that can precisely detect and isolate simultaneously occurring actuator faults and sensor faults for uncertain linear stochastic systems. The proposed approach is an observer based FDI scheme where a discontinuous observer is used for residual generation. The main highlights of the proposed FDI scheme are • Multiple actuator and sensor faults are considered • Proposed technique involves reconstruction of sensor faults and therefore this approach can be used for sensor fault identification and reconstruction of true outputs • There are no constraints on system uncertainties, both matched and mismatched uncertainties are considered • Present scheme can be easily extended to nonlinear systems by considering Lipschitz continuous affine nonlinear terms with known Lipschitz constant [13], [15] The structure of this paper is as follows. A detailed formulation of the observer based FDI scheme is first given. Afterwards, the results from numerical simulations and the concluding remarks are given in sections III and IV, respectively.
II. O BSERVER - BASED FAULT D ETECTION F ILTER Let (Ω, F, {Ft }t≥t0 , P) denotes a complete filtered probability space. Consider an nth -order stochastic system of the following form: ˙ 1 (t) = A11 X1 (t) + A12 X2 (t) + W1 (t) X ˙ 2 (t) = A21 X1 (t) + A22 X2 (t) + Bu(t) + W2 (t) X Y1 (t) = C11 X1 (t) + C12 X2 (t) + V 1 (t) Y2 (t) = C21 X1 (t) + C22 X2 (t) + ye (t) + V 2 (t)
(1)
where W1 (t) and W2 (t) denote stochastic disturbances and V 1 (t) and V 2 (t) indicate measurement noises. The state vectors, X1 (t) and X2 (t), are of dimensions, X1 (t) ∈ ℜn−r and X2 (t) ∈ ℜr , respectively. The true state matrices, A11 ∈ ℜ(n−r)×(n−r) , A12 ∈ ℜ(n−r)×r , A21 ∈ ℜr×(n−r) , A22 ∈ ℜr×r , and the control distribution matrix B ∈ ℜr×r , are assumed to be unknown. The desired input signal is denoted as ud (t) and ue (t) indicates the error in applied control, u(t), due to actuator faults, i.e., u(t) = ud (t) + ue (t)
(2)
The stochastic measurement vectors, Y1 (t) and Y2 (t), are of dimensions, Y1 (t) ∈ ℜm1 and Y2 (t) ∈ ℜm2 , respectively. The output matrices, C11 , C12 , C21 and C22, Tare assumed to be known. The measurement noise, V T1 (t) V T2 (t) = V (t) ∈ ℜm , is assumed to be zero-mean Gaussian white noise process, i.e., V (t) ∼ N 0, Rδ(τ ) . The vector, ye (t), indicates sensor failures and is modeled as y˙ e (t) = f (ye (t), t),
ye (t0 ) = 0
(3)
where f (·) is an unknown operator. Stochastic external disturbance T T = W(t) ∈ ℜn is modeled as a linear system W1 (t) W2T (t) driven by a Gaussian white noise process, i.e., ˙ W(t) = L(W(t), t) + W (t),
W(t0 ) = 0
(4)
where L(·) is an unknown linear operator and W (t) ∈ ℜn , is a zero mean Gaussian white noise process, i.e., W (t) ∼ N 0, Qδ(τ ) .
Assumption 1. Assume the sensor faults are not instantaneous and therefore there exist a known conservative upper bound on f (ye (t), t) such that |f (ye (t), t)| ≤ σ(t), ∀t ≥ t0 where | · | denotes the Euclidean norm. The external disturbance, W(t), is mean square bounded [16], [17], i.e., h i sup E W(t)WT (t) ≤ K t≥t0
where K is a constant matrix whose elements are finite. The assumed (known) model of the plant in (1) is given as x˙ m1 (t) = Am11 xm1 (t) + Am12 xm2 (t) x˙ m2 (t) = Am21 xm1 (t) + Am22 xm2 (t) + Bm ud (t)
(5)
Define the model-error vectors D 1 (t) ∈ ℜn−r and D 2 (t) ∈ ℜr as D 1 (t) = ∆A11 X1 (t) + ∆A12 X2 (t) + W1 (t) D 2 (t) = ∆A21 X1 (t) + ∆A22 X2 (t) + ∆Bud (t) + W2 (t)
(6)
where ∆A11 = A11 − Am11 , ∆A12 = A12 − Am12 , ∆A21 = A21 − Am21 , ∆A22 = A22 − Am22 , and ∆B = B − Bm . Assumption 2. Given the system parameter uncertainties are bounded and the system states are bounded in mean square sense, an upper bound on the model error vector D(t) can be obtained as P (|D(t)| ≤ µ ¯(t)) = 1,
∀t ≥ t0
That is, |D(t)| is almost surely (a.s.) upper bounded by µ ¯(t) for all t ≥ t0 . Now the plant dynamics in (1) can be written in-terms of known parameters as ˙ 1 (t) = Am11 X1 (t) + Am12 X2 (t) + D 1 (t) X ˙ 2 (t) = Am21 X1 (t) + Am22 X2 (t) + Bm ud (t) + D 2 (t) + Bue (t) X Re-parameterize X2 (t) as X2 (t) = αX2α (t) + βX2β (t)
(7)
where X2α (t) ∈ ℜr , X2β (t) ∈ ℜr , α and β are user selected scalar ˙ 2 (t) can be written as X ˙ 2 (t) = αX ˙ 2α (t) + parameters. Now X ˙ ˙ ˙ β X2β (t). Select X2α (t) and X2β (t) as ˙ 2α (t) = 1 Am21 X1 (t) + Am22 X2α (t) + 1 D 2 (t) X α α 1 1 ˙ X2β (t) = Am22 X2β (t) + Bm ud (t) + Bue (t) β β
(8)
Remark 1. One of the main challenges in the design of observer based FDI scheme is the presence of coupled system uncertainties and actuator faults [15]. In the presence of coupled system uncertainties and actuator faults, it is difficult to design an observer that yields measurement residual which is only sensitive to the actuator faults. Notice that the re-parametrization of X2 (t) given in (7) allows decoupling of system uncertainties and actuator faults as shown in (8).
After appending the sensor error extended system can be written as ˙ 1 (t) X Am11 αAm12 1 X ˙ Am21 Am22 2α (t) = α X ˙ 2 (t) 0 0 β 0 0 y˙ e (t) I 0 0 0 + 1 B ud (t) + 0 β m 0 0
0 1 I α
0 0
m
β
0
I
h(·)
˙ Z(t) = F Z(t) + G3 ud (t) + G1 D 1 (t) + G2 D 2 (t) + G3 ζ(t) (9) + G4 h(·) Am11 αAm12 βAm12 0 1 A Am22 0 0 and where F = α m21 0 0 Am22 0 0 0 0 Aye I 0 0 0 1 0 α I 0 0 G , G1 G2 G3 G4 = 1 0 0 B 0 m β 0 0 0 I C11 αC12 βC12 0 Let H = , the measurement equations can C21 αC22 βC22 I be rewritten as Y(t) = HZ(t) + V (t). Now the system in (1) can be written as the following dynamically equivalent form ˙ Z(t) = F Z(t) + G3 ud (t) + G1 D 1 (t) + G2 D 2 (t) + G3 ζ(t) + G4 h(·) Y(t) = HZ(t) + V (t)
(10)
Remark 2. Even though the above representation of the plant is a nonminimal realization, the observability of the extended system may be obtained by making appropriate changes to the state matrix, F , and the corresponding changes to D 1 (t), D 2 (t), and h(·). Consider the following partition of G1 as n − r column vectors, G2 as r column vectors, G3 as r column vectors, and G4 as m2 column vectors as shown below G1 = g11 g12 . . . g1(n−r) G2 = g21 g22 . . . g2r G3 = g31 g32 . . . g3r G4 = g41 g42 . . . g4m2 Also consider the individual elements of the vectors ζ(t), D(t), and h(·), i.e., h1 (·) D1 (t) ζ1 (t) ζ(t) = ... , D(t) = ... and h(·) = ... hm2 (·) ζr (t) Dn (t)
Now the extended system in (9) can be written in summation form as ˙ Z(t) =F Z(t) +
n−r X
g1i Di (t) +
r X
k=1
g3k ζk (t) +
r X
g2j Dn−r+j (t)
j=1
i=1
+ ∀t ≥ t0
0 X1 (t) 0 X2α (t) 0 X2β (t) ye (t) Aye 0 0 D 1 (t) 0 0 D 2 (t) 1 B 0 ζ(t)
βAm12 0 Am22 0
where h(·) = f (·) − Aye ye and Aye ∈ ℜm2 ×m2 is a user selected T Hurwitz matrix. Let Z(t) = XT1 (t) XT2α (t) XT2β (t) yeT (t) , now the above extended system can be rewritten as
Assumption 3. Assume there exists a bounded vector ζ(t) ∈ ℜr such that Bue (t) = Bm ζ(t), i.e., −1 ζ(t) = Bm Bue (t) and |ζ(t)| ≤ ξ(t),
dynamics given in (3), the
m2 X l=1
g4l hl (·) + G3 ud (t)
(11)
Define G 1 , G1 G2 , G 2 , G3 G4 , and η T (t) , T T ζ (t) hT (·) . Now (11) may be rewritten as ˙ Z(t) = F Z(t) + G3 ud (t) +
n X
r+m2
G 1i Di (t) +
i=1
X
G 2j ηj (t) (12)
j=1
where G 1k and G 2k are the kth column vectors of G 1 and G 2 matrices, respectively. Now ℓ = 1, 2, . . . , r + 1 observers of the following from are considered If ℓ ≤ r
h i ℓ b˙ (t) = F Z b ℓ (t) + Lℓ Y(t) − H Z b ℓ (t) + G3 ud (t)+ Z n X
G 1i µℓi (t)
+
i=1
ℓ−1 X
r+m2
G 2j νjℓ (t)
+
j=1
If ℓ = r + 1
X
j=ℓ+1
h
i
ℓ b˙ (t) = F Z b ℓ (t) + Lℓ Y(t) − H Z b ℓ (t) + G3 ud (t)+ Z n X
(13)
G 2j νjℓ (t)
r+m2
G 1i µℓi (t)
+
i=1
X
(14)
i=1
G 1i Di (t) −
µℓi (t)
j=1
i
r+m2
+
X
h
G 2j ηj (t) −
j=1
νjℓ (t)
i=1
X j=1
h
G 2j ηj (t) − νjℓ (t)
i
e ℓ (t) = Z e ℓ (t) − Z e ℓm (t), now subtracting (18) from (16) yields Let Z h n i ℓ h i X e ℓ (t) = e (t) + dZ F − Lℓ H Z G 1i Di (t) − µℓi (t) +
i
X
j=1
h
G 2j ηj (t) −
νjℓ (t)
i
(19)
dt
Given next is an approach for the selection of the observer gain Lℓ and the observer inputs µℓ (t) and ν ℓ (t) corresponding to the ℓth observer based on the stochastic Lyapunov approach. Since the only information regarding the true observer error is in the form of measurement residual, one do not have full access to the e ℓ (t), i.e., one only has access to Y e ℓ (t) = H Z e ℓ (t). Based signal Z ℓ e (t) can be written as on (19), dY h i ℓ ℓ e e (t)+ dY (t) = H F − Lℓ H Z n X i=1
h
HG 1i Di (t) −
µℓi (t)
i
r+m2
+
X
j=1
h
HG 2j ηj (t) −
νjℓ (t)
i
dt
Based on Assumptions 1 and 3, define an upper bound on η(t) as |η(t)| ≤ ν¯(t),
∀t ≥ t0
(15)
It is important to note that the solution to the stochastic differential equation given in (15) cannot be based on the ordinary mean square calculus because the integral involved in the solution depends on V (t), which is of unbounded variation, i.e., E V (t)V T (t + τ ) = Rδ(τ ). For the treatment of this class of problems, the stochastic differential equation can be rewritten in Itˆ o form as [18], [19] h n i h i X e ℓ (t) + e ℓ (t) = G 1i Di (t) − µℓi (t) + dZ F − Lℓ H Z r+m2
Notice that the almost sure stability of the observer error is impossible due to the persistently acting measurement noise B(t). Therefore it is desirable for the observer error corresponding to the e ℓ (t), to have a dynamics that follows ℓth observer, Z h i e ℓm (t) = e ℓm (t) dt + Lℓ dB(t) dZ F − Lℓ H Z (18)
r+m2
Equations (13) and (14) correspond to the typical observer model. The observer gain Lℓ and the observer inputs µℓ (t) and ν ℓ (t) corresponding to the ℓth observer are selected so that the generated residual obtained from observers given in (13) is asymptotically stable if there is no fault in the ℓth actuator and the residual obtained from the observer given in (14) is asymptotically stable despite any actuator or sensor fault occurrences. e ℓ (t) = Z(t) − Z b ℓ (t). After Define the observer error as Z subtracting (14) from (11), the observer error dynamics can be written as h i ℓ e˙ (t) = F − Lℓ H Z e ℓ (t) − Lℓ V (t)+ Z h
e ℓ (t) is asymptotically stable Definition 1. The stochastic process Z with probability 1, or almost surely asymptotically stable, if e ℓ (t) → 0 as t → ∞ = 1 P Z (17)
i=1
G 2j νjℓ (t)
where Lℓ ∈ ℜ(n+r+m2 )×m is the observer gain correspondth ing inputs are denoted as, ℓ to the ℓ ℓ observer.TThe ℓobserver r+m 2 ν1 (t) . . . ν(r+m2 ) (t) , ν (t) ∈ ℜ , and ℓ T µ1 (t) . . . µℓn (t) , µℓ (t) ∈ ℜn , ∀ℓ = 1, 2, . . . , r + 1
n X
and therefore the stability of the observer error dynamics given in (16) is depicted either as moment stability or stability in probabilistic sense. The stability in probabilistic sense is usually known as almost sure (a.s.) stability and it is defined as follows [16]:
(16)
dt − Lℓ dB(t)
where the zero-mean Gaussian white noise V (t) is written as the increments of stationary Wiener process with zero-mean and the correlation of increments h i E {B(τ ) − B(ζ)} {B(τ ) − B(ζ)}T = R|τ − ζ| Details on stochastic Itˆ o calculus can be found in [19]. The observer e ℓ (t), is a stochastic process error corresponding to the ℓth observer, Z
Theorem 1. Given the Assumptions 1, 2, and 3, the individual fault detection filters given in Eq. (13) guarantee almost sure asymptotic e ℓ (t) if there is no fault occurrence in the ℓth actuator stability of Y and the fault detection filter given in Eq. (14) guarantees almost e r+1 (t) despite any actuator or sensor sure asymptotic stability of Y fault occurrences, if the observer gain Lℓ corresponding to the ℓth observer is selected so that the following matrix Lyapunov inequality is satisfied h iT F − Lℓ H H T HP ℓ H T H+ (20) h i H T HP ℓ H T H F − Lℓ H + Qℓ ≤ 0 and the observer inputs corresponding to the ℓth observer are selected as T ℓ ℓ T ℓ e Y (t) HP H HG 1i µ ¯(t), µi (t) = sgn (21) ∀i = 1, . . . , n T e ℓ (t) HP ℓ H T HG 2i ν¯(t), Y νjℓ (t) = sgn (22) ∀j = 1, . . . , r + m2
where P ℓ ∈ ℜ(n+r+m2 )×(n+r+m2 ) and Qℓ ∈ ℜ(n+r+m2 )×(n+r+m2 ) are positive definite symmetric matrices and sgn{·} denotes the signum function or the sign function. Proof: Construct a Lyapunov function candidate of the form ℓ T ℓ e e e ℓ (t). Now using the Itˆo forV (Y (t)) = Y (t) HP ℓ H T Y e ℓ (t)) can be calculated as mula [19], dV (Y ( ℓ T h iT ℓ e e (t) dV (Y (t)) = Y F − Lℓ H H T HP ℓ H T H+ ℓ
h
T
ℓ
H HP H H F − L H
i
n ℓ T h i X e (t) HP ℓ H T H +2 Y G 1i Di (t) − µℓi (t)
ℓ
e (t) +2 Y
T
i=1 r+m2
ℓ
T
HP H H
X
h
G 2j ηj (t) −
j=1
νjℓ (t)
i
e ℓ (t) Y
)
and the assumed system matrices are 0 0 1 0 3.19 −0.31 , Am12 = , Bm = Am11 = 0 0 0 1 −0.1 2.01 −0.99 0 −1.13 0 , Am22 = Am21 = 0 −2.98 0 −1.18
dt
After substituting (20), LV (y˜ℓ ) can be written as T e ℓ (t) Qℓ y e ℓ (t) LV (y˜ℓ ) ≤ − y +2
n X i=1 r+m2
+2
e ℓ (t) y
X j=1
T
e ℓ (t) y
h i HP ℓ H T HG 1i Di (t) − µℓi (t)
T
h i HP ℓ H T HG 2j ηj (t) − νjℓ (t)
where the operator L{·} acting on V (x, t) is defined as LV (x, t) = lim
dt→0
1 E dV (X(t), t)|X(t) = x dt
(23)
2
X
j=1
Thus
(
e ℓ (t) | y
T
e ℓ (t) y
T
Note that D 1 = 0 and D 2 (t) is given as
HP ℓ H T HG 2j ηj (t)−
HP ℓ H T HG 2j |¯ ν (t)
D 2 (t) = ∆A21 X1 (t) + ∆A22 X2 (t) + ∆Bud (t) + W2 (t)
)
T LV (y˜ℓ (t)) ≤ − y˜ℓ (t) Qℓ y˜ℓ (t)
W21 (t) = −W21 (t) + W1 (t) W22 (t) = −W22 (t) + W2 (t)
The measurement noise, V (t) , V (t) ∈ R6 , is assumed to be zeromean Gaussian white noise process with h i E V (t)V T (t + τ ) = 10−2 × I6×6 δ(τ )
)
T e ℓ (t) HP ℓ H T HG 1i |¯ | y µ(t) + r+m2
= The systemT output matrices are given Tas C11 I 0 0 I , C = , C = 2×2 2×2 2×2 2×2 12 21 0.95 1.7 0.43 −2.31 , C22 = . For simulation −2.4 1.54 1.3 0.43 purposes, the external disturbance is modeled as W1 = 0 and W2 (t) = [W21 (t) W22 (t)]T is given as
where [W1 (t) W2 (t)]T = W (t) is zero-mean Gaussian white noise process with h i E W (t)W T (t + τ ) = 10−2 × I2×2 δ(τ )
Substituting (21) and (22) yields T e ℓ (t) Qℓ y e ℓ (t)+ LV (y˜ℓ ) ≤ − y ( n T X e ℓ (t) HP ℓ H T HG 1i Di (t)− y 2 i=1
III. N UMERICAL S IMULATIONS Numerical simulation results are presented in this section to validate the efficiency of the proposed FDI scheme. Consider a stochastic system of the form given in (1) where the true system matrices are given as 0 0 1 0 −1.3 0.01 A11 = , A12 = , A21 = , 0 0 0 1 −0.12 −1.8 −0.9 −0.011 2.4 −0.23 A22 = , B= 0.3 −3.4 −0.11 1.5
where the desired control input ud (t) is given in Fig. 1. For the re-parametrization of the system states, the constants α and β are selected as α = β = 1. Note that the two possible sensors faults are 10
(24)
Therefore the ℓ = (r + 1)th observer in (14) is almost surely asymptotically stable despite the occurrence of any actuator or sensor faults. Based on the given proof one could easily make the argument that if there is no fault occurrence in the ℓth actuator, where 1 ≤ ℓ ≤ r, then the ℓth observer given in (13) is almost surely asymptotically stable. e ℓ (t), will indicate a fault occurrence in Thus any observed residual y th the ℓ actuator. Any observed residual in the ℓth observer given in (13), where 1 ≤ ℓ ≤ r, indicates a fault occurrence in the ℓth actuator. Based on the observability condition one could easily show that the estimated ˆ e (t), asymptotically or the observer generated sensor error terms, y ˆ e (t) obtained approaches the true sensor error, ye (t). Therefore y
ud ud1 2
5
0
ud (t)
T
from the observer given in (14) can be directly used for sensor fault ˆ e (t) = 0, then there is no sensor fault and detection. That is, if y if yˆei (t) 6= 0, then the nonzero yˆei (t) indicates a fault occurrence ˆ e (t) from the measured in the ith sensor. Moreover, by subtracting y output yields the true system output.
−5
−10
−15
−20
−25
0
5
Fig. 1.
10
15
20
Time(sec)
25
30
Desired Control Input
associated with the fifth and sixth outputs. Two fault scenarios are considered here and the details on the fault scenarios are
For simulation purposes the upper bounds on D 2 (t), ζ(t), and h(·) are selected as |D21 (t)| ≤ 10 |D22 (t)| ≤ 20 |ζ1 (t)| ≤ 4 |ζ2 (t)| ≤ 3 |h1 (t)| ≤ 2 |h2 (t)| ≤ 2 20
2.5
D1 D2
15
ζ(t)
1.5
5
0
ue1 ue2
ye1 ye2
−0.5
1
0 0.5
−5 3
ζ1 ζ1
2
10
D 2 (t)
1) Fault Scenario I: For the first fault scenario, the faults are associated with the second actuator and the fifth output sensor. The actuator fault occurs at thirty seconds (sec) and the sensor fault occurs at sixty-five sec. Given in Fig. 2 are the ue (t) and ye (t) corresponding to the first fault scenario. 2) Fault Scenario II: For the second fault scenario considered, it is assumed that the faults are associated with the first actuator, the fifth and the sixth output sensors. The fault associated with the first actuator occurs at thirty sec. The fifth and sixth output sensor faults occur at sixty-five sec and eighty-five sec, respectively. Given in Fig. 3 are the ue (t) and ye (t) corresponding to the second fault scenario.
−10
0
20
2.5
40
60
80
100
Time(sec)
0
120
0
20
40
60
80
Time(sec)
100
120
−1
(a) Model Error
ye (t)
ue (t)
2 −1.5
1.5
(b) Input Error
−2 0
h2 h1
−2.5 −0.1
1 −3
−0.2
0.5 −3.5
0
20
40
60
80
Time(sec)
100
120
−4
−0.3
0
(a) Actuator Faults
20
40
60
80
Time(sec)
100
h(t)
0
−0.4
120
−0.5 −0.6
(b) Sensor Faults
−0.7 −0.8
Fig. 2.
Fault Scenario I
−0.9
0
20
40
60
80
Time(sec)
100
120
(c) Output Error Rates 0
8
ue1 ue2
−0.5
−2
−2.5 −3
0.1
2
0.5
Obv1 Obv2
0.08 0
−3.5
y ˆe1 y ˆe2
0 −0.5
0.06
0
20
40
60
80
Time(sec)
100
120
−4
0
20
40
60
80
Time(sec)
100
120
˜ y(t)
−2
ˆ e (t) y
−1
−4 −4.5 −5
Fault Scenario I: Error Vectors
4
ye (t)
ue (t)
−1.5
Fig. 4.
ye1 ye2
6
−1
0.04
−1.5
0.02
−2
−2.5
0
−3
(a) Actuator Faults
−0.02
(b) Sensor Faults
−3.5 −0.04
Fig. 3.
−0.06
Fault Scenario II
Note that for the system considered here, there are two actuators and therefore three different observers are designed. For both fault scenarios considered, Ay is selected as Ay = 02×2 , P ℓ is selected as P ℓ = 10−2 × I, ∀ℓ ∈ {1, 2, 3}. The observer gain is calculated as L1 = L2 = L3 = 49.9299 0.3753 0.1847 −0.3075 0.1847 −0.3075 −47.3614 120.0819
−0.2986 49.8876 0.0195 0.3204 0.0195 0.3204 −85.1196 −76.9340
−0.2402 −0.3033 24.9291 0.5508 24.9291 0.5508 −21.7003 −64.9756
0.7390 −0.2856 −0.3715 24.6784 −0.3715 24.6784 115.5758 −21.7627
−2.5331 0.7403 1.0916 1.2925 1.0916 1.2925 49.6501 0.8362
−0.3157 −1.7170 −0.7319 1.9531 −0.7319 1.9531 −0.7843 49.8110
The extended output matrix H and the matrix G can be calculated as 02×2 02×2 02×2 C11 C12 C12 04×2 02×2 02×2 I , G = 2×2 H= C21 C22 C22 I2×2 02×2 Bm 02×2 02×2 02×2 I2×2
Details on the results obtained for both fault scenarios are given next. 1) Fault Scenario I: Given in Fig. 4 are the true error vectors, D 2 (t) = ∆A21 X1 (t) + ∆A22 X2 (t) + ∆Bud (t) + W2 (t), ζ(t) = −1 Bm Bue (t), and h(·) = f (·) corresponding to the first fault scenario.
−4 0
20
40
60
80
Time(sec) (a) Residual
Fig. 5.
100
120
−4.5
0
20
40
60
80
Time(sec)
100
120
(b) Estimated Sensor Faults
Fault Scenario I: Observer Residual and Estimated Sensor Faults
Given in Fig. 5 are the generated residual and the estimated sensor errors corresponding to the first fault scenario. Figure 5(a) contains the measurement residual generated for observer one and two. The first two kinks in the residual are due to the start of input application that occurs around five seconds and the leveling-off the input to its steady state value around twenty seconds. Notice the jump in observer two residual around thirty seconds due to the fault occurrence in the second actuator. Figure 5(b) contains the estimated sensor errors obtained from the third observer. Note that the estimated sensor error is similar to the true sensor error given in Fig. 2(b). 2) Fault Scenario II: Given in Fig. 6 are the error vectors, D 2 (t), ζ(t), and h(·) corresponding to the second fault scenario. The upper bounds on error vectors used here are the same upper bounds used for the first fault scenario. Given in Fig. 7 are the generated residual and the estimated sensor errors corresponding to the second fault scenario. Figure 7(a) contains the measurement residual generated for observer one and two. Notice the sudden increase in observer one residual around thirty seconds due to the fault occurrence in the first actuator. Figure 7(b) contains the estimated sensor errors obtained from the third observer. Note that the estimated sensor errors are similar to the true sensor errors given in Fig. 3(b).
20
0.5
D1 D2
can be directly used for sensor fault detection, isolation and identification. Moreover, by subtracting the estimated sensor errors from the measured outputs, true system outputs can be generated. The simulation results reveal clear indication of actuator faults despite the presence of matched system uncertainties and external disturbances. Moreover, the estimated sensor errors are identical to the true sensor error regardless of the measurement noise present.
ζ1 ζ1
0
15 −0.5 −1
ζ(t)
D 2 (t)
10
−1.5
5
0
−2
−2.5 −3
−5 −3.5 −10
0
20
40
60
80
100
Time(sec)
−4
120
0
20
(a) Model Error
40
60
80
Time(sec)
100
120
(b) Input Error
1.5
h2 h1
h(t)
1
0.5
0
−0.5
−1
0
20
40
60
80
Time(sec)
100
120
(c) Output Error Rates Fig. 6.
Fault Scenario II: Error Vectors
1.4
8
1.2
6
Obv1 Obv2
4
ˆ e (t) y
˜ y(t)
1 0.8 0.6 0.4
2
0
−2
0.2 0
−4
−0.2
−6
0
20
40
60
80
Time(sec) (a) Residual
Fig. 7.
100
120
y ˆe1 y ˆe2 0
20
40
60
80
Time(sec)
100
120
(b) Estimated Sensor Faults
Fault Scenario II: Observer Residual and Estimated Sensor Faults
IV. C ONCLUSION Robust actuator/sensor fault detection is a challenging problem due to the effects of modeling errors, system process noise, and measurement noise. This manuscript outlines the formulation of a robust fault detection and isolation scheme that can precisely detect and isolate simultaneously occurring actuator faults and sensor faults for uncertain linear stochastic systems. The given robust fault detection scheme would be able to distinguish between model uncertainties and actuator failure and therefore eliminate the problem of false alarms. The presented approach involves precise reconstruction of sensor faults and therefore this approach can be used for sensor fault identification and the reconstruction of true outputs from faulty sensor outputs. The proposed approach is an observer based fault detection and isolation scheme where a discontinuous observer is used for residual generation. The proposed approach assume conservative upper bounds on the system uncertainties, the actuator faults, and the sensor fault rates. A bank of discontinuous observers is designed for fault detection and isolation scheme where the number of observers is based on the number of actuators. The observer gain and the discontinuous observer inputs are selected so that the observed residual is almost surely asymptotically stable if there is no actuator fault occurrence. As a result, any observed residual would indicate a fault occurrence in the corresponding actuator. In addition to the bank of observers designed for actuator fault detection, a robust discontinuous observer is designed so that the estimated or the observer generated sensor error terms asymptotically approaches the true sensor error. Therefore, the sensor error estimates obtained from the robust observer
R EFERENCES [1] J. Gertler, Fault Detection and Diagnosis in Engineering Systems. New York: Marcel Dekker, 1998. [2] A. Willsky, “A survey of design methods for failure detection in dynamic systems,” Automatica, vol. 12, no. 6, pp. 601–611, November 1976. [3] R. V. Beard, “Failure accommodation in linear systems through selfreorganization,” Ph.D. Thesis, Massachusetts Institute of Technology, 1971. [4] H. L. Jones, “Failure detection in linear systems,” Ph.D. Thesis, Massachusetts Institute of Technology, 1973. [5] R. L. Chen, “Failure detection filters for robust analytical redundancy,” Ph.D. Thesis, University of California, Los Angeles, 2000. [6] B. Shafai, C. Pi, and S. Nork, “Robust fault detection using proportional integral observers,” Proceedings of the 5th Biannual World Automation Congress, vol. 14, pp. 515–520, 2002. [7] T. Menke and P. Maybeck, “Sensor/actuator failure detection in the vista f-16 by multiple model adaptive estimation,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 31, no. 4, pp. 1218–1229, October 1995. [8] R. Isermann, “Process fault detection based on modeling and estimation methods: A survey,” Automatica, vol. 20, no. 4, pp. 387–404, July 1984. [9] C. Edwards, S. K. Spurgeon, and R. J. Patton, “Sliding mode observers for fault detection and isolation,” Automatica, vol. 36, no. 4, pp. 541– 553, April 2000. [10] C. P. Tan and C. Edwards, “Sliding mode observers for detection and reconstruction of sensor faults,” Automatica, vol. 38, no. 10, pp. 1815– 1821, October 2002. [11] ——, “Sliding mode observers for robust detection and reconstruction of actuator and sensor faults,” International Journal of Robust and Nonlinear Control, vol. 13, no. 5, pp. 443–463, April 2003. [12] B. Jiang, M. Staroswiecki, and V. Cocquempot, “Fault estimation in nonlinear uncertain systems using robust sliding-mode observers,” Control Theory and Applications, IEE Proceedings of, vol. 151, no. 1, pp. 29–37, January 2004. [13] X.-G. Yan and C. Edwards, “Robust sliding mode observer-based actuator fault detection and isolation for a class of nonlinear systems,” International Journal of Systems Science, vol. 39, no. 4, pp. 349–359, April 2008. [14] W. Chen and M. Saif, “A sliding mode observer-based strategy for fault detection, isolation, and estimation in a class of lipschitz nonlinear systems,” International Journal of Systems Science, vol. 38, no. 12, pp. 943–955, January 2007. [15] ——, “Observer-based strategies for actuator fault detection, isolation, and estimation in a class of nonlinear systems,” Control Theory & Applications, IET, vol. 38, no. 12, pp. 943–955, January 2007. [16] H. J. Kushner, Stochastic Stability and Control. New York, NY: Academic Press, 1967, ch. 2. [17] T. T. Soong, Random Differential Equations in Science and Engineering. New York, NY: Academic Press, 1973, ch. 4-5. [18] T. T. Soong and M. Grigoriu, Random Vibration of Mechanical and Structural Systems. Englewood Cliffs, NJ: Prentice Hall, 1993, ch. 4. [19] M. Grigoriu, Stochastic Calculus. Boston, MA: Birkh¨auser, 2002, ch. 1-5.
I. I NTRODUCTION Faults are deviations from the normal behavior of the plant or its instrumentation and they can be categorized into: i) additive process faults, ii) multiplicative process faults, iii) sensor faults, and iv) actuator faults. There exist several fault monitoring procedures which can be used to recognize and distinguish different types of faults [1]. These fault monitoring procedures can be categorized into: i) fault detection, ii) fault isolation, and iii) fault identification. A survey on design methods for fault detection is given in [2]. Most of the existing FDI (fault detection and isolation) schemes are based on measurement residual generation. Generated residual is used to facilitate the decision making procedures involved in FDI. The basic difference between most FDI schemes is the underlying residual generation methods. Few examples of different FDI schemes are the observer based Fault Detection Filters [3]–[5], Kalman filter based Proportional Integral Observers [6], Multiple Model Adaptive Estimators [7], and system identification methods [8]. In this manuscript two types of faults are of concern, i.e., actuator faults and sensor faults. The FDI scheme considered here is the observer based approach. Discontinuous observers such as the sliding mode observers have been successfully used in FDI context [9]. Design of sliding mode observers for detection and reconstruction of actuator and sensor faults is presented in [10] and [11], respectively. The precise reconstruction of faults proposed in [10] assumes the absence of uncertainty. The FDI approaches presented in [11] and [12] introduce an approximate fault reconstruction scheme by minimizing the error between the true fault signal and the reconstructed fault signal. In [13], an FDI scheme for a class of nonlinear uncertain systems is presented by introducing limitations on the structure of the uncertainty. It is important to notice that the precise reconstruction of fault is generally not available in the presence of uncertainty. While the FDI schemes presented in [10]–[13] involves reconstruction of faults, sliding mode observer based FDI approach presented in [14] is based on the measurement residual generation. Though the FDI scheme presented in [14] assumes precise knowledge of system dynamics, a similar FDI scheme which is robust to mismatched uncertainties is presented in [15]. The residual generation presented in [14] and [15] is based on the sliding mode observer scheme where the observer maintain the sliding motion in the presence of mismatched uncertainties, but when fault occurs, the sliding motion is broken and a residual is generated which contains information regarding the fault. J. George is with Dept. of Mechanical & Aerospace Engineering, University at Buffalo, SUNY, Buffalo, NY-14260 [email protected] I. M. Gregory is with NASA Langley Research Center, Hampton, VA-23681
[email protected]
Though there are numerous literatures on fault detection and identification, very little work is done on developing robust fault detection schemes. Robust actuator/sensor fault detection is a challenging problem due to system modeling errors, external disturbances, and measurement noise. Most of the current fault detection algorithms are model based and they tent to induce false alarms when the plant dynamics differ from the assumed model. This manuscript presents a robust FDI scheme that can precisely detect and isolate simultaneously occurring actuator faults and sensor faults for uncertain linear stochastic systems. The proposed approach is an observer based FDI scheme where a discontinuous observer is used for residual generation. The main highlights of the proposed FDI scheme are • Multiple actuator and sensor faults are considered • Proposed technique involves reconstruction of sensor faults and therefore this approach can be used for sensor fault identification and reconstruction of true outputs • There are no constraints on system uncertainties, both matched and mismatched uncertainties are considered • Present scheme can be easily extended to nonlinear systems by considering Lipschitz continuous affine nonlinear terms with known Lipschitz constant [13], [15] The structure of this paper is as follows. A detailed formulation of the observer based FDI scheme is first given. Afterwards, the results from numerical simulations and the concluding remarks are given in sections III and IV, respectively.
II. O BSERVER - BASED FAULT D ETECTION F ILTER Let (Ω, F, {Ft }t≥t0 , P) denotes a complete filtered probability space. Consider an nth -order stochastic system of the following form: ˙ 1 (t) = A11 X1 (t) + A12 X2 (t) + W1 (t) X ˙ 2 (t) = A21 X1 (t) + A22 X2 (t) + Bu(t) + W2 (t) X Y1 (t) = C11 X1 (t) + C12 X2 (t) + V 1 (t) Y2 (t) = C21 X1 (t) + C22 X2 (t) + ye (t) + V 2 (t)
(1)
where W1 (t) and W2 (t) denote stochastic disturbances and V 1 (t) and V 2 (t) indicate measurement noises. The state vectors, X1 (t) and X2 (t), are of dimensions, X1 (t) ∈ ℜn−r and X2 (t) ∈ ℜr , respectively. The true state matrices, A11 ∈ ℜ(n−r)×(n−r) , A12 ∈ ℜ(n−r)×r , A21 ∈ ℜr×(n−r) , A22 ∈ ℜr×r , and the control distribution matrix B ∈ ℜr×r , are assumed to be unknown. The desired input signal is denoted as ud (t) and ue (t) indicates the error in applied control, u(t), due to actuator faults, i.e., u(t) = ud (t) + ue (t)
(2)
The stochastic measurement vectors, Y1 (t) and Y2 (t), are of dimensions, Y1 (t) ∈ ℜm1 and Y2 (t) ∈ ℜm2 , respectively. The output matrices, C11 , C12 , C21 and C22, Tare assumed to be known. The measurement noise, V T1 (t) V T2 (t) = V (t) ∈ ℜm , is assumed to be zero-mean Gaussian white noise process, i.e., V (t) ∼ N 0, Rδ(τ ) . The vector, ye (t), indicates sensor failures and is modeled as y˙ e (t) = f (ye (t), t),
ye (t0 ) = 0
(3)
where f (·) is an unknown operator. Stochastic external disturbance T T = W(t) ∈ ℜn is modeled as a linear system W1 (t) W2T (t) driven by a Gaussian white noise process, i.e., ˙ W(t) = L(W(t), t) + W (t),
W(t0 ) = 0
(4)
where L(·) is an unknown linear operator and W (t) ∈ ℜn , is a zero mean Gaussian white noise process, i.e., W (t) ∼ N 0, Qδ(τ ) .
Assumption 1. Assume the sensor faults are not instantaneous and therefore there exist a known conservative upper bound on f (ye (t), t) such that |f (ye (t), t)| ≤ σ(t), ∀t ≥ t0 where | · | denotes the Euclidean norm. The external disturbance, W(t), is mean square bounded [16], [17], i.e., h i sup E W(t)WT (t) ≤ K t≥t0
where K is a constant matrix whose elements are finite. The assumed (known) model of the plant in (1) is given as x˙ m1 (t) = Am11 xm1 (t) + Am12 xm2 (t) x˙ m2 (t) = Am21 xm1 (t) + Am22 xm2 (t) + Bm ud (t)
(5)
Define the model-error vectors D 1 (t) ∈ ℜn−r and D 2 (t) ∈ ℜr as D 1 (t) = ∆A11 X1 (t) + ∆A12 X2 (t) + W1 (t) D 2 (t) = ∆A21 X1 (t) + ∆A22 X2 (t) + ∆Bud (t) + W2 (t)
(6)
where ∆A11 = A11 − Am11 , ∆A12 = A12 − Am12 , ∆A21 = A21 − Am21 , ∆A22 = A22 − Am22 , and ∆B = B − Bm . Assumption 2. Given the system parameter uncertainties are bounded and the system states are bounded in mean square sense, an upper bound on the model error vector D(t) can be obtained as P (|D(t)| ≤ µ ¯(t)) = 1,
∀t ≥ t0
That is, |D(t)| is almost surely (a.s.) upper bounded by µ ¯(t) for all t ≥ t0 . Now the plant dynamics in (1) can be written in-terms of known parameters as ˙ 1 (t) = Am11 X1 (t) + Am12 X2 (t) + D 1 (t) X ˙ 2 (t) = Am21 X1 (t) + Am22 X2 (t) + Bm ud (t) + D 2 (t) + Bue (t) X Re-parameterize X2 (t) as X2 (t) = αX2α (t) + βX2β (t)
(7)
where X2α (t) ∈ ℜr , X2β (t) ∈ ℜr , α and β are user selected scalar ˙ 2 (t) can be written as X ˙ 2 (t) = αX ˙ 2α (t) + parameters. Now X ˙ ˙ ˙ β X2β (t). Select X2α (t) and X2β (t) as ˙ 2α (t) = 1 Am21 X1 (t) + Am22 X2α (t) + 1 D 2 (t) X α α 1 1 ˙ X2β (t) = Am22 X2β (t) + Bm ud (t) + Bue (t) β β
(8)
Remark 1. One of the main challenges in the design of observer based FDI scheme is the presence of coupled system uncertainties and actuator faults [15]. In the presence of coupled system uncertainties and actuator faults, it is difficult to design an observer that yields measurement residual which is only sensitive to the actuator faults. Notice that the re-parametrization of X2 (t) given in (7) allows decoupling of system uncertainties and actuator faults as shown in (8).
After appending the sensor error extended system can be written as ˙ 1 (t) X Am11 αAm12 1 X ˙ Am21 Am22 2α (t) = α X ˙ 2 (t) 0 0 β 0 0 y˙ e (t) I 0 0 0 + 1 B ud (t) + 0 β m 0 0
0 1 I α
0 0
m
β
0
I
h(·)
˙ Z(t) = F Z(t) + G3 ud (t) + G1 D 1 (t) + G2 D 2 (t) + G3 ζ(t) (9) + G4 h(·) Am11 αAm12 βAm12 0 1 A Am22 0 0 and where F = α m21 0 0 Am22 0 0 0 0 Aye I 0 0 0 1 0 α I 0 0 G , G1 G2 G3 G4 = 1 0 0 B 0 m β 0 0 0 I C11 αC12 βC12 0 Let H = , the measurement equations can C21 αC22 βC22 I be rewritten as Y(t) = HZ(t) + V (t). Now the system in (1) can be written as the following dynamically equivalent form ˙ Z(t) = F Z(t) + G3 ud (t) + G1 D 1 (t) + G2 D 2 (t) + G3 ζ(t) + G4 h(·) Y(t) = HZ(t) + V (t)
(10)
Remark 2. Even though the above representation of the plant is a nonminimal realization, the observability of the extended system may be obtained by making appropriate changes to the state matrix, F , and the corresponding changes to D 1 (t), D 2 (t), and h(·). Consider the following partition of G1 as n − r column vectors, G2 as r column vectors, G3 as r column vectors, and G4 as m2 column vectors as shown below G1 = g11 g12 . . . g1(n−r) G2 = g21 g22 . . . g2r G3 = g31 g32 . . . g3r G4 = g41 g42 . . . g4m2 Also consider the individual elements of the vectors ζ(t), D(t), and h(·), i.e., h1 (·) D1 (t) ζ1 (t) ζ(t) = ... , D(t) = ... and h(·) = ... hm2 (·) ζr (t) Dn (t)
Now the extended system in (9) can be written in summation form as ˙ Z(t) =F Z(t) +
n−r X
g1i Di (t) +
r X
k=1
g3k ζk (t) +
r X
g2j Dn−r+j (t)
j=1
i=1
+ ∀t ≥ t0
0 X1 (t) 0 X2α (t) 0 X2β (t) ye (t) Aye 0 0 D 1 (t) 0 0 D 2 (t) 1 B 0 ζ(t)
βAm12 0 Am22 0
where h(·) = f (·) − Aye ye and Aye ∈ ℜm2 ×m2 is a user selected T Hurwitz matrix. Let Z(t) = XT1 (t) XT2α (t) XT2β (t) yeT (t) , now the above extended system can be rewritten as
Assumption 3. Assume there exists a bounded vector ζ(t) ∈ ℜr such that Bue (t) = Bm ζ(t), i.e., −1 ζ(t) = Bm Bue (t) and |ζ(t)| ≤ ξ(t),
dynamics given in (3), the
m2 X l=1
g4l hl (·) + G3 ud (t)
(11)
Define G 1 , G1 G2 , G 2 , G3 G4 , and η T (t) , T T ζ (t) hT (·) . Now (11) may be rewritten as ˙ Z(t) = F Z(t) + G3 ud (t) +
n X
r+m2
G 1i Di (t) +
i=1
X
G 2j ηj (t) (12)
j=1
where G 1k and G 2k are the kth column vectors of G 1 and G 2 matrices, respectively. Now ℓ = 1, 2, . . . , r + 1 observers of the following from are considered If ℓ ≤ r
h i ℓ b˙ (t) = F Z b ℓ (t) + Lℓ Y(t) − H Z b ℓ (t) + G3 ud (t)+ Z n X
G 1i µℓi (t)
+
i=1
ℓ−1 X
r+m2
G 2j νjℓ (t)
+
j=1
If ℓ = r + 1
X
j=ℓ+1
h
i
ℓ b˙ (t) = F Z b ℓ (t) + Lℓ Y(t) − H Z b ℓ (t) + G3 ud (t)+ Z n X
(13)
G 2j νjℓ (t)
r+m2
G 1i µℓi (t)
+
i=1
X
(14)
i=1
G 1i Di (t) −
µℓi (t)
j=1
i
r+m2
+
X
h
G 2j ηj (t) −
j=1
νjℓ (t)
i=1
X j=1
h
G 2j ηj (t) − νjℓ (t)
i
e ℓ (t) = Z e ℓ (t) − Z e ℓm (t), now subtracting (18) from (16) yields Let Z h n i ℓ h i X e ℓ (t) = e (t) + dZ F − Lℓ H Z G 1i Di (t) − µℓi (t) +
i
X
j=1
h
G 2j ηj (t) −
νjℓ (t)
i
(19)
dt
Given next is an approach for the selection of the observer gain Lℓ and the observer inputs µℓ (t) and ν ℓ (t) corresponding to the ℓth observer based on the stochastic Lyapunov approach. Since the only information regarding the true observer error is in the form of measurement residual, one do not have full access to the e ℓ (t), i.e., one only has access to Y e ℓ (t) = H Z e ℓ (t). Based signal Z ℓ e (t) can be written as on (19), dY h i ℓ ℓ e e (t)+ dY (t) = H F − Lℓ H Z n X i=1
h
HG 1i Di (t) −
µℓi (t)
i
r+m2
+
X
j=1
h
HG 2j ηj (t) −
νjℓ (t)
i
dt
Based on Assumptions 1 and 3, define an upper bound on η(t) as |η(t)| ≤ ν¯(t),
∀t ≥ t0
(15)
It is important to note that the solution to the stochastic differential equation given in (15) cannot be based on the ordinary mean square calculus because the integral involved in the solution depends on V (t), which is of unbounded variation, i.e., E V (t)V T (t + τ ) = Rδ(τ ). For the treatment of this class of problems, the stochastic differential equation can be rewritten in Itˆ o form as [18], [19] h n i h i X e ℓ (t) + e ℓ (t) = G 1i Di (t) − µℓi (t) + dZ F − Lℓ H Z r+m2
Notice that the almost sure stability of the observer error is impossible due to the persistently acting measurement noise B(t). Therefore it is desirable for the observer error corresponding to the e ℓ (t), to have a dynamics that follows ℓth observer, Z h i e ℓm (t) = e ℓm (t) dt + Lℓ dB(t) dZ F − Lℓ H Z (18)
r+m2
Equations (13) and (14) correspond to the typical observer model. The observer gain Lℓ and the observer inputs µℓ (t) and ν ℓ (t) corresponding to the ℓth observer are selected so that the generated residual obtained from observers given in (13) is asymptotically stable if there is no fault in the ℓth actuator and the residual obtained from the observer given in (14) is asymptotically stable despite any actuator or sensor fault occurrences. e ℓ (t) = Z(t) − Z b ℓ (t). After Define the observer error as Z subtracting (14) from (11), the observer error dynamics can be written as h i ℓ e˙ (t) = F − Lℓ H Z e ℓ (t) − Lℓ V (t)+ Z h
e ℓ (t) is asymptotically stable Definition 1. The stochastic process Z with probability 1, or almost surely asymptotically stable, if e ℓ (t) → 0 as t → ∞ = 1 P Z (17)
i=1
G 2j νjℓ (t)
where Lℓ ∈ ℜ(n+r+m2 )×m is the observer gain correspondth ing inputs are denoted as, ℓ to the ℓ ℓ observer.TThe ℓobserver r+m 2 ν1 (t) . . . ν(r+m2 ) (t) , ν (t) ∈ ℜ , and ℓ T µ1 (t) . . . µℓn (t) , µℓ (t) ∈ ℜn , ∀ℓ = 1, 2, . . . , r + 1
n X
and therefore the stability of the observer error dynamics given in (16) is depicted either as moment stability or stability in probabilistic sense. The stability in probabilistic sense is usually known as almost sure (a.s.) stability and it is defined as follows [16]:
(16)
dt − Lℓ dB(t)
where the zero-mean Gaussian white noise V (t) is written as the increments of stationary Wiener process with zero-mean and the correlation of increments h i E {B(τ ) − B(ζ)} {B(τ ) − B(ζ)}T = R|τ − ζ| Details on stochastic Itˆ o calculus can be found in [19]. The observer e ℓ (t), is a stochastic process error corresponding to the ℓth observer, Z
Theorem 1. Given the Assumptions 1, 2, and 3, the individual fault detection filters given in Eq. (13) guarantee almost sure asymptotic e ℓ (t) if there is no fault occurrence in the ℓth actuator stability of Y and the fault detection filter given in Eq. (14) guarantees almost e r+1 (t) despite any actuator or sensor sure asymptotic stability of Y fault occurrences, if the observer gain Lℓ corresponding to the ℓth observer is selected so that the following matrix Lyapunov inequality is satisfied h iT F − Lℓ H H T HP ℓ H T H+ (20) h i H T HP ℓ H T H F − Lℓ H + Qℓ ≤ 0 and the observer inputs corresponding to the ℓth observer are selected as T ℓ ℓ T ℓ e Y (t) HP H HG 1i µ ¯(t), µi (t) = sgn (21) ∀i = 1, . . . , n T e ℓ (t) HP ℓ H T HG 2i ν¯(t), Y νjℓ (t) = sgn (22) ∀j = 1, . . . , r + m2
where P ℓ ∈ ℜ(n+r+m2 )×(n+r+m2 ) and Qℓ ∈ ℜ(n+r+m2 )×(n+r+m2 ) are positive definite symmetric matrices and sgn{·} denotes the signum function or the sign function. Proof: Construct a Lyapunov function candidate of the form ℓ T ℓ e e e ℓ (t). Now using the Itˆo forV (Y (t)) = Y (t) HP ℓ H T Y e ℓ (t)) can be calculated as mula [19], dV (Y ( ℓ T h iT ℓ e e (t) dV (Y (t)) = Y F − Lℓ H H T HP ℓ H T H+ ℓ
h
T
ℓ
H HP H H F − L H
i
n ℓ T h i X e (t) HP ℓ H T H +2 Y G 1i Di (t) − µℓi (t)
ℓ
e (t) +2 Y
T
i=1 r+m2
ℓ
T
HP H H
X
h
G 2j ηj (t) −
j=1
νjℓ (t)
i
e ℓ (t) Y
)
and the assumed system matrices are 0 0 1 0 3.19 −0.31 , Am12 = , Bm = Am11 = 0 0 0 1 −0.1 2.01 −0.99 0 −1.13 0 , Am22 = Am21 = 0 −2.98 0 −1.18
dt
After substituting (20), LV (y˜ℓ ) can be written as T e ℓ (t) Qℓ y e ℓ (t) LV (y˜ℓ ) ≤ − y +2
n X i=1 r+m2
+2
e ℓ (t) y
X j=1
T
e ℓ (t) y
h i HP ℓ H T HG 1i Di (t) − µℓi (t)
T
h i HP ℓ H T HG 2j ηj (t) − νjℓ (t)
where the operator L{·} acting on V (x, t) is defined as LV (x, t) = lim
dt→0
1 E dV (X(t), t)|X(t) = x dt
(23)
2
X
j=1
Thus
(
e ℓ (t) | y
T
e ℓ (t) y
T
Note that D 1 = 0 and D 2 (t) is given as
HP ℓ H T HG 2j ηj (t)−
HP ℓ H T HG 2j |¯ ν (t)
D 2 (t) = ∆A21 X1 (t) + ∆A22 X2 (t) + ∆Bud (t) + W2 (t)
)
T LV (y˜ℓ (t)) ≤ − y˜ℓ (t) Qℓ y˜ℓ (t)
W21 (t) = −W21 (t) + W1 (t) W22 (t) = −W22 (t) + W2 (t)
The measurement noise, V (t) , V (t) ∈ R6 , is assumed to be zeromean Gaussian white noise process with h i E V (t)V T (t + τ ) = 10−2 × I6×6 δ(τ )
)
T e ℓ (t) HP ℓ H T HG 1i |¯ | y µ(t) + r+m2
= The systemT output matrices are given Tas C11 I 0 0 I , C = , C = 2×2 2×2 2×2 2×2 12 21 0.95 1.7 0.43 −2.31 , C22 = . For simulation −2.4 1.54 1.3 0.43 purposes, the external disturbance is modeled as W1 = 0 and W2 (t) = [W21 (t) W22 (t)]T is given as
where [W1 (t) W2 (t)]T = W (t) is zero-mean Gaussian white noise process with h i E W (t)W T (t + τ ) = 10−2 × I2×2 δ(τ )
Substituting (21) and (22) yields T e ℓ (t) Qℓ y e ℓ (t)+ LV (y˜ℓ ) ≤ − y ( n T X e ℓ (t) HP ℓ H T HG 1i Di (t)− y 2 i=1
III. N UMERICAL S IMULATIONS Numerical simulation results are presented in this section to validate the efficiency of the proposed FDI scheme. Consider a stochastic system of the form given in (1) where the true system matrices are given as 0 0 1 0 −1.3 0.01 A11 = , A12 = , A21 = , 0 0 0 1 −0.12 −1.8 −0.9 −0.011 2.4 −0.23 A22 = , B= 0.3 −3.4 −0.11 1.5
where the desired control input ud (t) is given in Fig. 1. For the re-parametrization of the system states, the constants α and β are selected as α = β = 1. Note that the two possible sensors faults are 10
(24)
Therefore the ℓ = (r + 1)th observer in (14) is almost surely asymptotically stable despite the occurrence of any actuator or sensor faults. Based on the given proof one could easily make the argument that if there is no fault occurrence in the ℓth actuator, where 1 ≤ ℓ ≤ r, then the ℓth observer given in (13) is almost surely asymptotically stable. e ℓ (t), will indicate a fault occurrence in Thus any observed residual y th the ℓ actuator. Any observed residual in the ℓth observer given in (13), where 1 ≤ ℓ ≤ r, indicates a fault occurrence in the ℓth actuator. Based on the observability condition one could easily show that the estimated ˆ e (t), asymptotically or the observer generated sensor error terms, y ˆ e (t) obtained approaches the true sensor error, ye (t). Therefore y
ud ud1 2
5
0
ud (t)
T
from the observer given in (14) can be directly used for sensor fault ˆ e (t) = 0, then there is no sensor fault and detection. That is, if y if yˆei (t) 6= 0, then the nonzero yˆei (t) indicates a fault occurrence ˆ e (t) from the measured in the ith sensor. Moreover, by subtracting y output yields the true system output.
−5
−10
−15
−20
−25
0
5
Fig. 1.
10
15
20
Time(sec)
25
30
Desired Control Input
associated with the fifth and sixth outputs. Two fault scenarios are considered here and the details on the fault scenarios are
For simulation purposes the upper bounds on D 2 (t), ζ(t), and h(·) are selected as |D21 (t)| ≤ 10 |D22 (t)| ≤ 20 |ζ1 (t)| ≤ 4 |ζ2 (t)| ≤ 3 |h1 (t)| ≤ 2 |h2 (t)| ≤ 2 20
2.5
D1 D2
15
ζ(t)
1.5
5
0
ue1 ue2
ye1 ye2
−0.5
1
0 0.5
−5 3
ζ1 ζ1
2
10
D 2 (t)
1) Fault Scenario I: For the first fault scenario, the faults are associated with the second actuator and the fifth output sensor. The actuator fault occurs at thirty seconds (sec) and the sensor fault occurs at sixty-five sec. Given in Fig. 2 are the ue (t) and ye (t) corresponding to the first fault scenario. 2) Fault Scenario II: For the second fault scenario considered, it is assumed that the faults are associated with the first actuator, the fifth and the sixth output sensors. The fault associated with the first actuator occurs at thirty sec. The fifth and sixth output sensor faults occur at sixty-five sec and eighty-five sec, respectively. Given in Fig. 3 are the ue (t) and ye (t) corresponding to the second fault scenario.
−10
0
20
2.5
40
60
80
100
Time(sec)
0
120
0
20
40
60
80
Time(sec)
100
120
−1
(a) Model Error
ye (t)
ue (t)
2 −1.5
1.5
(b) Input Error
−2 0
h2 h1
−2.5 −0.1
1 −3
−0.2
0.5 −3.5
0
20
40
60
80
Time(sec)
100
120
−4
−0.3
0
(a) Actuator Faults
20
40
60
80
Time(sec)
100
h(t)
0
−0.4
120
−0.5 −0.6
(b) Sensor Faults
−0.7 −0.8
Fig. 2.
Fault Scenario I
−0.9
0
20
40
60
80
Time(sec)
100
120
(c) Output Error Rates 0
8
ue1 ue2
−0.5
−2
−2.5 −3
0.1
2
0.5
Obv1 Obv2
0.08 0
−3.5
y ˆe1 y ˆe2
0 −0.5
0.06
0
20
40
60
80
Time(sec)
100
120
−4
0
20
40
60
80
Time(sec)
100
120
˜ y(t)
−2
ˆ e (t) y
−1
−4 −4.5 −5
Fault Scenario I: Error Vectors
4
ye (t)
ue (t)
−1.5
Fig. 4.
ye1 ye2
6
−1
0.04
−1.5
0.02
−2
−2.5
0
−3
(a) Actuator Faults
−0.02
(b) Sensor Faults
−3.5 −0.04
Fig. 3.
−0.06
Fault Scenario II
Note that for the system considered here, there are two actuators and therefore three different observers are designed. For both fault scenarios considered, Ay is selected as Ay = 02×2 , P ℓ is selected as P ℓ = 10−2 × I, ∀ℓ ∈ {1, 2, 3}. The observer gain is calculated as L1 = L2 = L3 = 49.9299 0.3753 0.1847 −0.3075 0.1847 −0.3075 −47.3614 120.0819
−0.2986 49.8876 0.0195 0.3204 0.0195 0.3204 −85.1196 −76.9340
−0.2402 −0.3033 24.9291 0.5508 24.9291 0.5508 −21.7003 −64.9756
0.7390 −0.2856 −0.3715 24.6784 −0.3715 24.6784 115.5758 −21.7627
−2.5331 0.7403 1.0916 1.2925 1.0916 1.2925 49.6501 0.8362
−0.3157 −1.7170 −0.7319 1.9531 −0.7319 1.9531 −0.7843 49.8110
The extended output matrix H and the matrix G can be calculated as 02×2 02×2 02×2 C11 C12 C12 04×2 02×2 02×2 I , G = 2×2 H= C21 C22 C22 I2×2 02×2 Bm 02×2 02×2 02×2 I2×2
Details on the results obtained for both fault scenarios are given next. 1) Fault Scenario I: Given in Fig. 4 are the true error vectors, D 2 (t) = ∆A21 X1 (t) + ∆A22 X2 (t) + ∆Bud (t) + W2 (t), ζ(t) = −1 Bm Bue (t), and h(·) = f (·) corresponding to the first fault scenario.
−4 0
20
40
60
80
Time(sec) (a) Residual
Fig. 5.
100
120
−4.5
0
20
40
60
80
Time(sec)
100
120
(b) Estimated Sensor Faults
Fault Scenario I: Observer Residual and Estimated Sensor Faults
Given in Fig. 5 are the generated residual and the estimated sensor errors corresponding to the first fault scenario. Figure 5(a) contains the measurement residual generated for observer one and two. The first two kinks in the residual are due to the start of input application that occurs around five seconds and the leveling-off the input to its steady state value around twenty seconds. Notice the jump in observer two residual around thirty seconds due to the fault occurrence in the second actuator. Figure 5(b) contains the estimated sensor errors obtained from the third observer. Note that the estimated sensor error is similar to the true sensor error given in Fig. 2(b). 2) Fault Scenario II: Given in Fig. 6 are the error vectors, D 2 (t), ζ(t), and h(·) corresponding to the second fault scenario. The upper bounds on error vectors used here are the same upper bounds used for the first fault scenario. Given in Fig. 7 are the generated residual and the estimated sensor errors corresponding to the second fault scenario. Figure 7(a) contains the measurement residual generated for observer one and two. Notice the sudden increase in observer one residual around thirty seconds due to the fault occurrence in the first actuator. Figure 7(b) contains the estimated sensor errors obtained from the third observer. Note that the estimated sensor errors are similar to the true sensor errors given in Fig. 3(b).
20
0.5
D1 D2
can be directly used for sensor fault detection, isolation and identification. Moreover, by subtracting the estimated sensor errors from the measured outputs, true system outputs can be generated. The simulation results reveal clear indication of actuator faults despite the presence of matched system uncertainties and external disturbances. Moreover, the estimated sensor errors are identical to the true sensor error regardless of the measurement noise present.
ζ1 ζ1
0
15 −0.5 −1
ζ(t)
D 2 (t)
10
−1.5
5
0
−2
−2.5 −3
−5 −3.5 −10
0
20
40
60
80
100
Time(sec)
−4
120
0
20
(a) Model Error
40
60
80
Time(sec)
100
120
(b) Input Error
1.5
h2 h1
h(t)
1
0.5
0
−0.5
−1
0
20
40
60
80
Time(sec)
100
120
(c) Output Error Rates Fig. 6.
Fault Scenario II: Error Vectors
1.4
8
1.2
6
Obv1 Obv2
4
ˆ e (t) y
˜ y(t)
1 0.8 0.6 0.4
2
0
−2
0.2 0
−4
−0.2
−6
0
20
40
60
80
Time(sec) (a) Residual
Fig. 7.
100
120
y ˆe1 y ˆe2 0
20
40
60
80
Time(sec)
100
120
(b) Estimated Sensor Faults
Fault Scenario II: Observer Residual and Estimated Sensor Faults
IV. C ONCLUSION Robust actuator/sensor fault detection is a challenging problem due to the effects of modeling errors, system process noise, and measurement noise. This manuscript outlines the formulation of a robust fault detection and isolation scheme that can precisely detect and isolate simultaneously occurring actuator faults and sensor faults for uncertain linear stochastic systems. The given robust fault detection scheme would be able to distinguish between model uncertainties and actuator failure and therefore eliminate the problem of false alarms. The presented approach involves precise reconstruction of sensor faults and therefore this approach can be used for sensor fault identification and the reconstruction of true outputs from faulty sensor outputs. The proposed approach is an observer based fault detection and isolation scheme where a discontinuous observer is used for residual generation. The proposed approach assume conservative upper bounds on the system uncertainties, the actuator faults, and the sensor fault rates. A bank of discontinuous observers is designed for fault detection and isolation scheme where the number of observers is based on the number of actuators. The observer gain and the discontinuous observer inputs are selected so that the observed residual is almost surely asymptotically stable if there is no actuator fault occurrence. As a result, any observed residual would indicate a fault occurrence in the corresponding actuator. In addition to the bank of observers designed for actuator fault detection, a robust discontinuous observer is designed so that the estimated or the observer generated sensor error terms asymptotically approaches the true sensor error. Therefore, the sensor error estimates obtained from the robust observer
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